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For an exterior circular grain with radiation-pressure coefficient $\beta$, set $q=\sqrt{1-\beta }$ and $ϵ=a/a_p-1$. Its next conjunction occurs after the synodic period, during which the planet's unwrapped mean longitude advances by $\Lambda =2\pi /[1-q(1+ϵ{)}^{-3/2}]$. A weak planetary scattering encounter changes this by $\delta \Lambda \simeq -3\pi q(1+ϵ{)}^{-5/2}\delta ϵ/[1-q(1+ϵ{)}^{-3/2}{]}^2$. Unwrapped longitude counts turns; reducing it modulo $2\pi$ before comparison would erase this measure of timing sensitivity.

If release imparts no velocity kick, a grain's initial Kepler orbit under reduced stellar attraction has $q=r_b$, $Q=r_b/(1-2\beta )$, and $e=\beta /(1-\beta )$ for $0\le \beta <1/2$. Its pericentre distance remains at the dust birth ring, while its apocentre distance grows with the radiation-pressure coefficient.

A zero of the angular drift in a cold single-grain-size dust tail produces a formally divergent number density per unit angle. The divergence is integrable when the exact local angle-to-age map is retained. A spread of launch velocities or radiation-pressure coefficients removes the singularity.

For a blackbody grain with geometric cross-section $\sigma$, mass $m$ and stellar-frame velocity $\mathbf{v}$, define the radiation-pressure coefficient $\beta =L_{\star }\sigma /(4\pi cG{M}_{\star }m)$. To first order in $v/c$, steady absorption and isotropic rest-frame reradiation giveThe relative incoming flux provides $1-\dot{r}/c$; photon reradiation carries stellar-frame momentum proportional to $\mathbf{v}/c$. Subtracting static radiation pressure gives radial drag $-2m\beta G{M}_{\star }\dot{r}/(c{r}^2)$ and transverse drag $-m\beta G{M}_{\star }{v}_{\theta }/(c{r}^2)$. Non-blackbody grains require their actual radiation-pressure efficiency.

The electromagnetic scattering solution for a sphere describes the transition between Rayleigh scattering and geometrical optics. Grain-size-dependent efficiencies can produce structure in the radiation-pressure coefficient.

Let $\mu =M_p/M_{\star }$, $q=\sqrt{1-\beta }$, and restrict first to $0\le \beta <1$. The radiation-pressure coefficient reduces the grain's central gravitational parameter to $G{M}_{\star }{q}^2$, whereas the planet's is approximately $G{M}_{\star }$. Their mean motions obey $n/n_p=q(1+ϵ{)}^{-3/2}$. Setting this ratio to $j/(j+1)$ gives the radiation-pressure shift of a mean-motion resonance:It lies interior to the planet exactly when $q(j+1)<j$, or $j>q/(1-q)$. Equality puts the frequency commensurability at the planet's radius. For $\beta =0$ every such commensurability is exterior.

There is a frame qualification in the printed encounter formula. The circular inertial speed of the grain at $a_p(1+{ϵ}_1)$ is $v_1=q{v}_p(1+{ϵ}_1{)}^{-1/2}$. Subtracting the planet's inertial velocity at conjunction gives the local planet-relative inertial speedThis is the velocity used in the question's two-body gravitational assist calculation. In a frame actually rotating about the star at $n_p$, the grain's tangential velocity is instead $v_1-n_p{a}_p(1+{ϵ}_1)$, with magnitude $v_p[1+{ϵ}_1-q(1-{ϵ}_1/2)]$. In particular its nonradiating shear is $3v_p{ϵ}_1/2$, rather than $v_p{ϵ}_1/2$. Thus the requested expression uses a local translating, inertial encounter convention despite the source's rotating-frame wording. The following derivation consistently uses that convention, and its coefficients are an encounter estimate rather than an exact rotating-frame result.

Take impact parameter $b\simeq a_p{ϵ}_1$ and a small gravitational scattering angle $\theta$. Expanding the supplied hyperbolic scattering relation gives $\theta \simeq 2G{M}_p/(b{v}_{\infty }^2)$. Initially the relative tangential velocity is $-v_{\infty }$; after the deflection it has tangential component $-v_{\infty }\cos \theta$. The relative speed is unchanged in the local hyperbolic Kepler orbit, but the stellar-frame kinetic energy changes. Taking the dot product with the planet's velocity givesThe specific orbital energy of the grain is $-G{M}_{\star }(1-\beta )/(2a)$. Comparing it immediately before and after the encounter at the same position givesConsequentlyThe encounter increases the grain's semi-major axis within this geometry. These steps assume ${ϵ}_1\ll 1$, $\theta \ll 1$, and an orbital kick small enough for the subsequent linearization.

Choose the initial conjunction longitude as zero. The synodic period is $2\pi /(n_p-n)$, and the planet's longitude accumulated by the next conjunction isThese are unwrapped longitudes, counting complete revolutions. The second uses the new mean orbital period while neglecting the encounter's immediate phase offset and the difference between true and mean conjunction. A comparison reduced modulo $2\pi$ could not implement the stated one-turn sensitivity test.

For $\beta =0$, $v_{\infty }\simeq v_p{ϵ}_1/2$, so $\delta ϵ\simeq 32{\mu }^2{ϵ}_1^{-5}$ and ${\lambda }_c\simeq 4\pi /(3ϵ)$. Conjunction-longitude sensitivity to an encounter givesRequiring this to exceed $2\pi$ gives ${ϵ}_1^7<64{\mu }^2/3$. At a high-index first-order mean-motion resonance, ${ϵ}_j=(1+1/j{)}^{2/3}-1\simeq 2/(3j)$, henceThe interpretation of the resonance-overlap encounter estimate is loss of coherent encounter phases: a small kick changes the next synodic encounter time by more than one planetary revolution. Resonant protection by repeatedly meeting at controlled orbital phases can then fail, allowing irregular kicks and diffusion. This sensitivity criterion motivates an unstable region; it is not a proof that every orbit or resonant phase in it must escape.

For a quantitative sketch with radiation, put $h=1-q(1-ϵ/2)$ and $d=1-q(1+ϵ{)}^{-3/2}$. Differentiating the conjunction formula and substituting the kick givesEvaluate this at the positive ${ϵ}_j$ and take $\mathcal{S}=1$ as the critical-index curve. At small $\beta$, increased encounter speed weakens planetary scattering, moving the threshold to larger $j$. As $\beta$ rises further, the radiation-pressure shift of a mean-motion resonance brings each commensurability closer to the planet, strengthening the close encounters. The curve turns over and approaches the exterior-limit boundary $j=q/(1-q)$. Larger planetary mass produces stronger kicks and a lower critical index. These competing effects are shown below; the curves retain the factors in $\mathcal{S}$, so their $\beta =0$ intercepts differ slightly from the leading high-$j$ formula. The boundary for disappearing exterior commensurabilities is geometric; divergence of the encounter approximation near $ϵ=0$ should not be read as a precise physical scattering rate.

For a spherical grain of density $\rho$, the radiation-pressure coefficient isThe $r^{-2}$ dependence of radiation pressure cancels that of stellar gravity. Here radiation-pressure efficiency includes absorption and the momentum-transfer part of scattering, averaged over the stellar spectrum. In the geometrical optics regime, $D$ exceeds the important stellar wavelengths, $Q_{\mathrm{pr}}$ is of order unity, and $\beta \propto D^{-1}$: cross-sectional area grows as $D^2$, whereas mass grows as $D^3$. Around the stellar wavelength, Mie scattering can produce a broad maximum and material-dependent structure. In the Rayleigh scattering regime, absorption gives $Q_{\mathrm{abs}}\propto D$, while scattering gives $Q_{\mathrm{sca}}\propto D^4$. Thus small absorbing grains approach an approximately constant $\beta$, whereas nearly transparent grains have $\beta \propto D^3$. A universal fall to zero at small $D$ is therefore inappropriate without specifying the optical properties. The sketch shows both possible small-grain limits; its vertical normalization is illustrative.

Assume release with negligible velocity relative to the planet, negligible planetary gravity after release, and a constant radiation-pressure coefficient. Put $\mu =G{M}_{\star }$ and ${\mu }_d=(1-\beta )\mu$. The inherited speed and specific angular momentum are $v_p=\sqrt{\mu /a_p}$ and $h=a_p{v}_p$. The new specific orbital energy isFor $0\le \beta <1/2$, the dust orbit released from a circular parent ring is an ellipse. Using ${\epsilon }_d=-{\mu }_d/(2a_d)$ and $h^2={\mu }_d{a}_d(1-e_d^2)$ givesFor $\beta >0$, release is at periapsis, since $a_d(1-e_d)=a_p$. At the radiation-pressure blowout threshold, $\beta =1/2$, the orbit is parabolic. For $1/2<\beta <1$ it is hyperbolic: the signed semimajor axis is negative and $e_d>1$. For $\beta \ge 1$ the central force ceases to be attractive, so the elliptical interpretation and the positive-eccentricity formula cannot be extended unchanged.

With ${\theta }_d$ measured from release, it is the true anomaly. The polar equation of a Kepler orbit and conservation of specific angular momentum giveDefine the signed dust-tail angular lag by $\Delta \theta ={\theta }_d-{\theta }_p$, with ${\theta }_p=n_{p}t$ and both angles unwrapped. For fixed ${\theta }_d$, expansion to first order in $\beta$ givesso $\Delta \theta =2\beta (\sin {\theta }_d-{\theta }_d)+O({\beta }^2{\theta }_d)$. The dust trails the planet, hence the signed lag is negative. Its slope is $-4\beta {\sin }^2({\theta }_d/2)$; it decreases monotonically, has horizontal tangents at successive release-direction passages, and has a sinusoidal ripple about $-2\beta {\theta }_d$. At the end of orbit $j$, ${\theta }_d=2\pi j$ and $\Delta \theta \simeq -4\pi \beta j$. At five orbits this is $-0.2\pi$ for $\beta =0.01$ and $-0.8\pi$ for $\beta =0.04$. The upper-$\beta$ sketch remains a first-order approximation; its accumulated error is $O({\beta }^2{\theta }_d)$.

For the subsequent positive-distance statements, set $s=-\Delta \theta ={\theta }_p-{\theta }_d\ge 0$. The mean motion of the grain isA full relative wrap, $s=2\pi$, therefore takes $t_{\mathrm{wrap}}\simeq \pi /(\beta n_p)$ after averaging over the orbital ripple. During this time Poynting–Robertson drag changes the semimajor axis byThe signed change is negative. This estimate treats $a_d\simeq a_p$ as fixed in the rate and computes the drag accumulated over the radiation-pressure wrap time. Since $v_p/c\ll 1$, drag causes little migration over that time. For an arbitrarily tiny $\beta$, however, this small migration can itself alter the relative phase appreciably: neglecting that feedback on the wrap time additionally requires $v_p/c\ll \beta$.

Rapid dust sublimation is a plausible sink near a hot planet: a grain can evaporate long before it completes a relative wrap. Destructive collisions can also remove visible grains. Poynting–Robertson drag alone, with the estimate above, does not explain immediate removal from a short tail. If the destruction time is $t_{\mathrm{loss}}$, a short tail requires roughly $2\beta n_p{t}_{\mathrm{loss}}\ll 2\pi$.

For a steady dust-tail continuity equation, assume a constant injection rate $\dot{N}$, grains with the same fixed $\beta$, negligible release-speed dispersion, and negligible destruction within the segment being calculated. A finite destruction lifetime can terminate that segment, or multiply its density by a survival probability. To first order,To obtain the explicit trigonometric profile, make the additional secular phase approximation for a dust tail, $s\simeq 2\beta {\theta }_d$, discarding the bounded $-2\beta \sin {\theta }_d$ term in the angle-to-age map while retaining the periodic instantaneous drift speed. ThenSince the square is even, the same written profile applies to the signed $\Delta \theta$ if number intervals are interpreted with positive orientation. The result is an approximate phase-substitution profile, not a uniformly valid consequence of $\beta \ll 1$. The consistent first-order description is instead parametric:In particular, immediately after release, $s\sim \beta {\theta }_d^3/3$ and $n\propto s^{-2/3}$, whereas the phase-substitution expression would give $s^{-2}$. Nor does $s\ll 2\pi$ by itself justify neglecting the orbital ripple. The formal enhancements where $\dot{s}=0$ are dust-tail caustics; a spread of radiation-pressure coefficients, finite release velocities, and destruction smooth them. With constant lifetime $\tau$, the parametric profile acquires $\exp [-t({\theta }_d)/\tau ]$. These qualifications state precisely the extra assumptions behind the explicit profile.

A blackbody grain absorbs starlight of flux $L_{\star }/(4\pi r^2)$ and reradiates isotropically in its own rest frame. Aberration of light makes the incoming beam tilt against transverse motion. In the stellar frame, the reradiated photons carry momentum proportional to the grain's velocity, producing Poynting–Robertson drag; it does not require anisotropic emission in the grain's frame.

Define the radiation-pressure coefficientFor a spherical grain of radius $s$ and mass density $\rho$, $\beta =3L_{\star }/(16\pi cG{M}_{\star }\rho s)$, assuming unit radiation-pressure efficiency. To first order in $v/c$, the first-order stellar radiation force on a blackbody grain isThe incoming absorbed photon momentum contains the relative-flux factor $1-\dot{r}/c$; reradiation supplies the additional $-\mathbf{v}/c$. Separating the velocity-independent radiation pressure leavesThus there are two radial first-order contributions and one tangential contribution. The approximation assumes $v\ll c$, steady absorption/reradiation and constant grain properties; a non-blackbody requires its actual optical efficiencies.

Treat each grain as an isolated compact sphere and take its wavelength-independent radiation-pressure efficiency to be $Q_{\mathrm{pr}}=1$. At distance $R$, its mass and radiation-force cross-section areThe inward gravitational force and outward radiation pressure force areThus the opacity per unit dust mass and the dust-grain Eddington limit areTo equal the Eddington luminosity based on electron-scattering opacity, ${\kappa }_d$ must equal ${\kappa }_{\mathrm{es}}$. ThereforeFor fully ionized hydrogen, ${\kappa }_{\mathrm{es}}={\sigma }_T/m_p\simeq 0.40\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$, and the stated ${\rho }_d=1\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$ givesUsing $0.34\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$ for a roughly solar-composition gas instead gives $2.2\mathrm{cm}$. This composition-dependent normalization should be stated because the problem does not fix it. For a different constant radiation-pressure efficiency, the critical radius is multiplied by $Q_{\mathrm{pr}}$.

At $L=L_{\mathrm{Edd},\mathrm{es}}$, the grain's radiation-pressure coefficient isConsequently, smaller grains experience a net outward force, larger grains remain gravitationally attracted, and critical-size grains have zero net radial force. Grains released at rest are therefore size-sorted into expelled and infalling populations. With an initial inward velocity, an outward force does not guarantee reversal before the grain reaches the hole; trajectories also depend on initial conditions. Gas drag and dust destruction have been omitted from this isolated-grain calculation. Ordinary interstellar grains are much smaller than a centimetre, so their direct radiative acceleration is very strong in this idealized model.

For a spherical grain of diameter $D$, density ${\rho }_d$, and radiation-pressure efficiency $Q_{\mathrm{pr}}$, comparison of the stellar momentum flux with Newton's law of universal gravitation gives the radiation-pressure coefficientFor $D$ much larger than the characteristic stellar wavelength, $Q_{\mathrm{pr}}\simeq 1$, so $\beta \propto D^{-1}$. When $D$ becomes comparable to the optical wavelength, diffraction and the grain's composition make $Q_{\mathrm{pr}}$ size dependent and $\beta$ reaches a broad maximum. Deep in the Rayleigh scattering regime, absorption can give $Q_{\mathrm{pr}}\propto D$ and hence nearly constant $\beta$, while scattering alone gives $Q_{\mathrm{pr}}\propto D^4$ and hence $\beta \propto D^3$. A realistic curve therefore rises roughly as $D^{-1}$ toward micron sizes, turns over near the stellar spectral peak, and falls or flattens for still smaller grains.

The impact gives fragments a spread of orbital energy and specific angular momentum. Their resulting spread of mean motion lets Keplerian shear stretch a compact dust clump into an arc and then a ring, while the spread of orbital frequencies causes phase mixing. Size-dependent radiation-pressure coefficients immediately place small grains on different eccentric or even radiation-pressure blowout orbits; Poynting–Robertson drag and stellar-wind drag then alter their orbits on longer timescales. Further collisions grind or disperse the clump, and planetary perturbations can accelerate mixing.

These processes depend strongly on $D_f$. Small grains have larger radiation-force-to-gravity ratios and generally shorter collisional or drag lifetimes, while larger fragments remain closer to the parent orbit but can preserve a velocity-dispersion-driven clump for longer. The lifetime also depends on collision location, ejection velocities, optical depth, and orbital radius. A universal fixed $\Delta t$ is therefore a useful population-model approximation, not a literal property of every collision; a size- and event-dependent lifetime distribution is more realistic.

The radiation-pressure coefficient reduces the dust grain's effective stellar gravitational parameter to $(1-\beta )G{M}_{\ast }$. At the exterior 5:4 mean-motion resonance, $n_d/n_p=4/5$, so mean motion givesThis is a first-order mean-motion resonance. Its leading disturbing-function term is therefore linear in the small orbital eccentricity:Thus its dimensionless strength is of order $(M_p/M_{\ast })e_d$, up to the Laplace-coefficient combination $f_{5:4}$, and its resonant argument varies slowly near commensurability.

A grain with radiation-pressure coefficient $\beta <1$ feels effective central gravitational parameter $G{M}_{\star }(1-\beta )$. Combining its mean motion with the planet's mean motion at an exterior-type $(j+1):j$ frequency commensurability gives the displayed semi-major axis. The commensurability lies geometrically exterior only for $j<\sqrt{1-\beta }/(1-\sqrt{1-\beta })$. Radiation changes the grain's attraction, so the same frequency ratio need not imply the usual radial ordering.

A direct transition from a solid phase to a gaseous phase. Heated interplanetary dust or a comet can lose material through this process, changing its size and radiation-pressure coefficient or destroying it before gravitational or drag-driven evolution is complete.

A synchrone is the locus, at one observation time, of grains released at the same time with different radiation-pressure coefficients.

A syndyne is the locus, at one observation time, of grains with the same radiation-pressure coefficient released at different times.